This document describes a co-evolving genetic algorithm called the filtering genetic algorithm (GA). It consists of two interacting GAs - GAfit and GAsch - and a filter. GAfit's population is evaluated using a modified function that incorporates the filter, which aims to direct the population away from areas of convergence found by GAsch. GAsch's population, consisting of schemas, is evaluated based on how well they cover strings in GAfit's population. The filter is updated based on the degree of convergence in GAfit's population to varying strengths influence GAfit's evaluation and drive it to new areas. The approach allows the algorithm to efficiently locate multiple optima in multimodal functions without getting stuck in local optim

1. t994 IEEE Symposium on
Emerging Technologies & Factory Automa1ion
Co-Evolving Genetic Algorithm with Filtered Evaluation Function
Hidenoti SAKANASHI and Yukinoti KAKAZU
Faculty of Engineering, Hokkaido University.
North- 13. West-8, Kita-ku , Sapporo, 060, Japan.
E-mail : {sakana. kakazu }@ bupe.bokudai.ac.jp
Abscracr - As a function optimizer or a search procedure. Genetic
search procedure applicable to a wide variety of problems, it is felt
that such remarkable weak points should be overcome . Hence ,
much research into GA-hard problems has been presented, and
many extended GAs have been proposed. Some of them focus on
maintaining adequate diversity of the strings in the population,
using sharing [4]. incest pre vention [3 ], crowding [10],
introducing ecological frameworks [2], and so on. Almost all of
them aim at solving the specific types of hard problem, and seem to
be over-sensitive to parameter senings.
Algorithms (GAs) are very powerful and have many advantages.
Fundamental research concerning the internal behavior of GAs has
highlighted their limitations as regards search performances for
what are called GA-hard problems. The reason for these difficulties
seems to be that GAs generate insufficient strategies for the
convergence of populations . To overcome this problem, an
extended GA. which we name the filtering-GA, that adopts the
concept of co-evolution, is proposed. It has two GAs , and the y
influence each other through their evaluation process.
3. CO-EVOLUTION
I. INTRODUCTION
In a static environment, agents, which have certain mechanisms for
improving their performance and can change its own structure, will
acquire stable structure and can show better performance. Then,
whether that perfonnance is optimal or not for given environment,
they wiU never change stable structure. Such structure is one of
local optima. On the other hand, in the dynamicaUy vary ing
environment, they may keep changing their struc ture to improve
performance and avoid sticking into local optima. If the
environment changes lo suppress their perfonnance impro vement,
moreover, it is diffi cu lt for the agents to improve their
performance, and it is expected that the structure acquired in such
en vironment is more adaptive and robust. Here, consider the
environmen l as problem, and agents as solutions or strategies for
obtaining them. It is reported that such system can solve [6] and is
called co-evolution system.
The co-evolutionary system has competitive re lationship
between its components. In above case, competitors have different
roles, e .g. , problem and solutions , but it is not only a type of coevolutionary system. Tournament evaluation in genetic
programming is adopted to reduce the computational load [!] .
There also is a system using two types of agents with different
aims, planning and scheduling, for solving the problem of
manufacturing [8].
In this study , there are one environment and two agents .
The environment is the objective function, and the agents are GAs.
Two GAs influence evaluation processes of opponents, but they
aren't competitors. One change the appearance of evaluation
function of opponent, and the population of the opponent is used in
evaluation of its population. Through this cyclic re lationship
between two GAs, the system can avoid local optima without
interfering optimization of the objective fW>Ction.
As a function optimizer or a search procedure , GAs are very
powerful, and can perform robust search through multi-point
search using the population. So they have been applied to many
fields {4]. Fundamental research concerning their internal beh.-ior,
howe ver. has re v e aled the lim itations as regards search
performance. This kind of problems singled out by suc h research
are known as GA -hard pro blems. Whether the y are GA-hard or
not, they may be optimized if there is some prior know ledge, but
this si tuation never o ccurs as it is not necessary to use GAs for
problems whose characteristics are already known.
The reason for difficu lties seems to be that, for the
convergence of po pulations, GAs generate insufficient strategies.
To overcome this problem, in this paper, we proposed an e. tended
GA whi ch adopts the concept of co-evolution. It has a mechanism
for controlling the convergence of populations . From the functiona1
point of view , the system consists of three components: the
observation of convergence. its force , and the repulsion against it.
To realize these functions, the system proposed in this paper uses
two GAs and a filter which stores the information showing the
trend of convergence . Two GAs influence each other through their
evaluation process.
2. GENETIC ALGORITHMS
In the simp le GA as a multi-point search procedure . the search
points of the objective function F, xi, are represented in the strings
of characters, S,, and are stored in the population , wh ich is
represented as [[S, ]]" in this paper. Here, supposing that the
strings are composed of the binary digit characters [0,1)', N is the
population size. Every string is mapped into the search space of the
objective function by the decoding function, D. They are tested in
the objective function and receive values using the evaluation
function, G. and
G(S,) = F(D{s,)) = F(x,)
(! )
On the basis of those values, the genetic operators act on the
population and produce the population of the ne xt generation.
Through repeated reproduction of the population, strings can
receive a larger value from the e valuation func tion. and the average
of those values will increase with each population.
The notion of the schema is important for analyzing the
behavior of GAs. It consists of the trinary characters [0.1.# 1. The
character # is called the don't care mark and matches both 0 and I.
That is to say, the schema is a set of perfectly matching strings. In
contrast to suings in the population that indicate points in the search
space. the schemata specify regions thereof. Clearly, schema with a
large order are likely to be e ffected by mutation. and schema with a
long defining length wiU tend to be influenced by crossover. This
fact is the essence of the schema theorem [7].
GA-hard problems !iteraUy mean types of problem which
simple GA can't easily sol ve. Since GAs are known as a robust
0-7603-2t14-6/941$4.00
© 1994 IEEE.
4. FlLTIRING GENETIC ALGORITHM
4.1 ConcepiUal D yTifl!1Jics of F i ltering-GA
GA-hard problems can be thought of as having a landscape with
many slatic or dynamic local peaks, and we now assume that the
objective is to maJtimize such a multimodal function . Without prior
knowledge for solving the problems to which they are being
applied. it is difficult for any search procedure to completely avoid
the local peaks. To suppress mistaken convergence on the local
peaks, the extended GA proposed in this paper, the filtering -GA,
repeats its searches many times. In this way, it is possible to repeat
the fast evolution of the population, and to discover many local
optima. Furthennore, if the system can restart from an unexplored
region, the search is expected to be more effective, and such search
will find the most promising regions sequentially until the search
space is completely explored or the system is terminated by an
external control.
To re liably locate many peaks in the landscape of a
multimodal objective Function in one computation, the filtering-GA
changes the effect of the objective function on the dynamics of the
GA. If the evaluation value around a local optimal area already
454
~-·
-
..
-- · . - --
----

2. discovered decreases, the population will move toward another
optimaJ area. In order _to realize ~ i s mec~s~ autonomously, we
propose the followmg operatmns: ftltenng, covering, and
extractl.on.
Whether the original string S is or isn't io the region represented by
the schema H , the new string S"" .. generated by the covering
operation is in this region. If the expected evaluation value of the
schema is large, it is quite possible that new strings generated by
tllis tnapping will get high evaluation values.
4.1 .I Filtering Operation
4 .1.3 Schema Extraction
The first operation of filtering is described here. The filter consists
of two sequences of length I, which correspond to the length of the
strings, and consist of real numbers, v" and v'. The jth element
of the sequence V' is represented hy v; Using t.his filter. the
The extracted schema is represented by H"' . The values of all its
p.ositions are defined by the following equations:
0, if z<l - P.., ,
(7)
1, ifz>P,u•
{ #, otherwise.
h =
't
string S, is evaluated by the modified evaluation function, as
follows:
G••• (S.)=G(S,)-
±•:•.
,.,
., _L:J
s'J
N
(2 )
(8)
•-
where s, is the value {0,1) of the jth position of string S,. As seen
whe re Pu, is the system parameter named the
in this equation, if v; is large. the strings which have values h (0 or
1) in lhejth position will receive a small evaluation value. Then, if
v;, in which j and h correspond to the local optima. has a large
value, the population will move away from these local optima and
IS expected to find another local optima. After it has converged on
one local optima, the population can be led away to another local
optima efficiently if the change of the filter is controlled properly.
Because the system doesn 't know what the local and global
optima are, it is impossible for the filter values to change when one
of the local optima is found. Therefore. the degree of convergence
ts used for deterrmmng the filter values. because the population
tends to converge on pealcs in the landscape of a problem whether
is calculated as:
they are optimal or not. Namely, all
threshold, 0.5 ~ Puo ~ 1, and z is the mean allele value at position
j . Eq. 7 is named the extract function.
v; =
v
;
*t,{
4.2 Schematic Structure of Filtering-GA
The filtering-GA consists of two GAs and a filter as shown in Fig.
2. OneGA is named GA,•. This is almost the same as a simple GA
except in its evaluation of strings. Systems which consist of several
GAs have already been proposed, such as the multi-level GA
[Grefenstette86] and so on. The two GAs in the filtering-GA malce
no hierarchy, and are not equal. They interact with each other and
evolve like a co-evolution system. The strings of GAm are
represented as S,w and the size of the population [[S,"Jt· is N,. Its
suings are evaluated through t.he filtering operation. The other GA
is named GAsch• and the members of its population are represented
in the form of schemata {0.1.#) . Every schema S,"" in the
v;
G~S,) xcomp(h,sJ}.
0, if h >' S· ,
h
comp(h ~s - ) = {
~'
v
1• ot erwtse.
extraction
(3 )
population [[S,"" J(· of GAsch is evaluated using the covering
(4)
operation. The population size of [[S,""]]"· is given by N,. The
filter is generated from GAsch and affects the evaluation of GA 50•
That is,
is the observed fllness of allele h at position j in the
current population. lf there are many strings whi c h ha,~ e value h in
the jth pos ition and their evaluation values are large. ': w ill be
large. The filter generated by these equations lowers the evaluation
values of strings which have value h in the jth position, in its
evaluation of the next generation by eq. 2 . As a result of this
procedure, the population moves away from the region where it has
converged (Fig. 1).
t~~~
"i"'jy,.t ~t ~
Fig. 2. Schematic diagram of filtering-GA.
4.3 Interaction among Two GAs and Objectivt Function
Fig. 1. Image of filtering operation.
4 .3 .I Evaluation Functions
4 .1.2 Covering Opera tion
The evaluation function of GAm• G,v, is given as the next
equation. and every soing S,"" of GAstr is evaluated a.s follows :
Th~
second ?peration of the filterin.g -GA is the covering operation.
Th1.s operation forces the populauon to converge in a particular
G,,(sw ) =G(S")-C,
reg10n [9] . Consider a schema Has a template of strings which
have windows at the position of the don't care marks #s. Then the
new string s~· is generated by putting this template on one string
S. This operation is formalized as follows:
s- = cover(H.S),
(5)
( )
_ _
1 , if hi "#,
5
6
' - hi, otherwtse.
±vf.
(9)
i•O
where G is given in eq. 1. s,~ is the value of the jth position of
S,'", v; is the jth value in V' of the filter, C, is the system
variable, and 0 s C, s 2 . Because the filter stores the tendency of
convergence, the population ofGA.a moves away from the region
of convergence. At each generation. the system records the
maximum and the average values calculated as G. If the best G
value discovered by GAm in a certain generation exceeds the
{s
155

3. average of all the G values obtained after P, generations ago. then it
would seem that GAsu is currently searching the most promising
region. Then, to weaken the influence of the filter on GA,~. C, is
reduced 2 x 1/ P, . The filter never affects GA,., when C, = 0. If
that condition is not satisfied. the search concerning the region
which GAm is currently searching seems to finish or that region is
not the most promising region. Then, tO stn:ngthen the influence of
the filter on GA,u. and to cause the population of GA 5u to move
away from that region, C, is increased l/ P,.. It is expected that the
influence of the filter will reject the convergence of GAsu when
C, =I, causing the power to avoid convergence when C, > I; and
when C, = 2. such avoidance power will be negatively coequal to
the convergence power of GA 50 with C, = 0. If it becomes C, < 0
or C, > 2 , C, is resettled at 0 or 2. P,. is named the history
number and is a parameter of the system.
Every schema of GAsch is evaluated by the following
equation:
C",(S;''•) = m;"'{c(cover(S,""
.s;•))}.
easily to be optimized. On the other hand, if a poor arrangement
(type- I) is used, the defining length of each building block is long
(type-2) , and then GAs can not easily converge to the optimal
solution. In both type, its global optima is 300.
Discontinuous Function is defined as follows :
kzK D,(S.),
F(S,J= j D 112
IK s;) '
<(
·
,
where
(10)
4.3 .2 Generation of Filter
The procedure for the generation of the filter is based on eq. 3 and
is modified as follows :
t:
••I
'
I_,,i=0.
,.,
(4-0·8 )
Otherwise,
I
K =300 and D,(S)= L :. , s,. It is deceptive function. To
optimize it, the schema of (0 ... 0# ... 11), in which all positions of left
half have Os. must be achieved. Its global optimum is 1.2 x K at
(0 ... 0 1. .. I) and its local optimum is K at (II... I ) .
The standard parameters used in the simulations are shown
in Table t. These parameters, for both the simple GA and the
filtering-GA, aren't adjusted for each problem. So, these two GAs
may be capable of exhibiting a higher performance than the results
shown here.
Fig. 3-a and 3-b repon the performances of the simple GA
and the filtering-GA on type-! and type-2 of 30-Bit, Order-Three
Deceptive Problem. These results were obtained by averaging over
5 independent runs. Although the population of the simple GA
completely converged onto a local optimum in both type, the
filtering-GA can find their optimal solutions.
Fig. 4 reports the performances of the simple GA and the
filtering-GA on Discontinuous Function. Although the population
of the simple GA completely converged onto a local optimum at
240, the filtering-GA failed optimization only once in 5 trials. This
failure was caused by the effect of the scheduling mechanism
introduced at eq. 9. The accurate acquisition of those schemata is
expected to be realized, if C, is changed in an adaptive way.
In both function, the filtering-GA discovered the local
optima as fast as the simple GA in the e"periments. Moreover,
premature 'on vergence never occurred, because it is able 10 avoid it
through using the filtering operation. Thus, the filtering-GA can
fast search local optima, using large selection pressure, without
convergence of the system.
This is the equation for obtaining the expected maximum evaluation
values of the schemata. The evaluation values obtained in this way
imply the possibility of evolution for the population in those
regions. Obviously, a large number of sampling points will ensure
accurate estimation. Therefore, all the strings of GAm arc utilized
in the evaluation of one schema of GAsch using the covering
operation.
_,
•
1 ~{a(s,"")
v,=N"- o(s.~')xcoml'h,s,,"")} ,
if
kE{O.I} . (II)
where o(H) is the order of the schema H. Based on the definition
of the schema, the don't care mark,#, matches both 0 and 1. but it
matches neither 0 nor I in the function comp(h,s) . The position of
#s in each schema never 3ffects the filter, because it has no
connection with the convergence of the search.
The population of GA..:h evolves and changes according to
eq. 10, and the filter generated from it varies, in each generation.
This means that the evaluation function of GAm given m eq. 9
changes at each generation. Under the evaluation function, which
changes drastically, the population doesn't evolve properly. Hence,
the average of all the filters generated after P,. generations ago is
used in eq. 9 . P, is the history number introduced at eq. 9.
6. CONCLUSION
This paper proposed an extended genetic algorithm , named the
fi ltering genetic algorithm (the filtering-GA ), as a multimodal
function optimizer. The filtering-GA is intended to solve GA-hard
multimodal functions, and to optimize GA-easy problems quickly.
It consists of two GAs and a filter. Two GAs influences each other
thorough their evaluation processes, and form a co-evolutionary
system. They suppress the convergence of oppenents, and avoid
sticking into local optima. The role of the filter is to smoothen the
interaction between the two GAs . Using the se three parts. the
filtering-GA can search widely throughout the search spaces of
problems, and discover many local optima sequentially. Its
conceptual and detailed definitions were given, and its functional
framework was discussed. In order to examine the search
performance of the filtering-GA, some experiments were carried
out, then it exhibited a higher performance and faster search than
the simple GA.
4.3 .3 Extraction nf Schema
To avoid system convergence, we introduce the third interaction
between the two GAs, named tl1e extraction. This operation is
used to estimate where GA,u is exploring in the search space, and
attracts the attention of GAscll to this region. In practice, the schema
which indicates such a region is extracted from the population of
GA, •. and is joined to the population of GAsch· To determine the
evaluation value Of the extracted Schema, CK>(Ho') , it is Settled as
max, G(S;'). Then the old schema. which has the lowest fitness
value in the population of GAsch. is replaced by this new extracted
schema H•'.
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5. COMPUTER SIMULATION
This section examines the search performance of the filtering·GA.
The objective functions used here are 30-Bit, Order-Three
Deceptive Problem (5] and Discontinuous FWiction {II].
30-Bit, Order-Three Deceptive Problem is created as the
sum of I 0 copies of the 3-bit function, where each copy is
associated with the next three bits. If all bits associated with a
particular subfunction are as close as possible, the function will
456

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Q)
"
OJ
>
c:
.Q
1ij
::>
iii
>
Q)
300
290
280
270
v;
Q)
.0
260
0
Strategy for Genetic AJgorithm s with Additional Genetic
400
600
generation
(a) Type- I .
800
1000
0
Q)
:::J
iii
>
c:
200
200
400
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generation
(b) Type-2.
BOO
I 000
300
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0
-~
Algorithms," Parallel Problem Solving from Nature , 2, pp.
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:::J
280
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>
Q)
270
1n
2l
260
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lnformlltion Systems. p. 51 7-520 (1993).
Fig. 3. Search performance of simple GA and filtering-GA on 30Bit, Order-Three Deceptive Problem.
Table 1. Parameter settings used in simulations.
Filtering-GA
Simple GA
1000
population size
30
string length
maximum generation 1000
on
elilist strategy
uniform
crossover type
0.8
crossover rate
0.03
mutation rate
history number
extraction threshold
GAstr
GAsch
30
30
1000
on
uniform
0.6
0. 03
30
Q)
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340
>
320
.Q
300
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280
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000
off
two-point
0. 8
0.05
;;;
iii
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Q)
1ii
Q)
260
240
.0
10
0.8
0
200
400
600
generation
BOO
1 000
Fig. 4. Search performance of simple GA and filtering-GA on
Discontinuous Function.
457